tomleslie

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These are answers submitted by tomleslie

In general the second argument to the indets() command has to be a Maple type - for a list of Maple types, see the help at ?type.

If (for some reason) you are trying to extract the operands of your expression, then you could use either of the methods shown in the attached


 

#
# The hard(?) way
#
  s1 := sqrt(2);
  s2 := log[2](3);
  s3 := surd(2, 3);
  map(op, indets(s1, op(0, s1)));
  map(op, indets(s2, op(0,s2)));
  map(op, indets(s3, op(0, s3)));
 

2^(1/2)

 

{2, 1/2}

 

ln(3)/ln(2)

 

{1/ln(2), ln(3)}

 

2^(1/3)

 

{2, 1/3}

(1)

#
# The easy(?) way
#
  op([0..-1], s1);
  op([0..-1], s2);
  op([0..-1], s3);

`^`, 2, 1/2

 

`*`, ln(3), 1/ln(2)

 

`^`, 2, 1/3

(2)

 


 

Download opStuff.mw

can be done with the attached.

Note that the numerator of your expression is real for any value of 'x'. However the denominator is complex for x<-4 or x>4.
Outside the range x=0..4, there are an infinite number of "solutions" of theform  0/a_complex_Number, which is obviously 0. However whether you consider this to be a "real" solution would seem to be a "philosophical" question

  restart;
  expr:=(-2*cos(x)^2+2*sin(x+(1/4)*Pi)^2-1)/sqrt(-x^2+4*x) = 0;
#
# "Basic" solution
#
  solve(expr, x);
#
# Now get "all" the solutions
#
  sols:=[solve(expr, x, allsolutions=true)];
#
# Produce "explicit" forms of the solutions
# over a limited range of integers - ie -10..10
# (which can be extended if necessary)
#
  ans:=`union`(seq( {seq( eval(sols[k],indets(sols[k], assignable)[]=j), j=-10..10)}, k=1..4));
#
# Check these solutions
#
  seq( eval(expr, x=j), j in ans);

(-2*cos(x)^2+2*sin(x+(1/4)*Pi)^2-1)/(-x^2+4*x)^(1/2) = 0

 

(1/2)*Pi, -(1/2)*Pi, (1/4)*Pi, -(3/4)*Pi

 

[(1/2)*Pi+2*Pi*_Z1, -(1/2)*Pi+2*Pi*_Z2, (1/4)*Pi+2*Pi*_Z3, -(3/4)*Pi+2*Pi*_Z3]

 

{-(83/4)*Pi, -(79/4)*Pi, -(75/4)*Pi, -(71/4)*Pi, -(67/4)*Pi, -(63/4)*Pi, -(59/4)*Pi, -(55/4)*Pi, -(51/4)*Pi, -(47/4)*Pi, -(43/4)*Pi, -(41/2)*Pi, -(39/2)*Pi, -(39/4)*Pi, -(37/2)*Pi, -(35/2)*Pi, -(35/4)*Pi, -(33/2)*Pi, -(31/2)*Pi, -(31/4)*Pi, -(29/2)*Pi, -(27/2)*Pi, -(27/4)*Pi, -(25/2)*Pi, -(23/2)*Pi, -(23/4)*Pi, -(21/2)*Pi, -(19/2)*Pi, -(19/4)*Pi, -(17/2)*Pi, -(15/2)*Pi, -(15/4)*Pi, -(13/2)*Pi, -(11/2)*Pi, -(11/4)*Pi, -(9/2)*Pi, -(7/2)*Pi, -(7/4)*Pi, -(5/2)*Pi, -(3/2)*Pi, -(3/4)*Pi, -(1/2)*Pi, (1/2)*Pi, (1/4)*Pi, (3/2)*Pi, (5/2)*Pi, (5/4)*Pi, (7/2)*Pi, (9/2)*Pi, (9/4)*Pi, (11/2)*Pi, (13/2)*Pi, (13/4)*Pi, (15/2)*Pi, (17/2)*Pi, (17/4)*Pi, (19/2)*Pi, (21/2)*Pi, (21/4)*Pi, (23/2)*Pi, (25/2)*Pi, (25/4)*Pi, (27/2)*Pi, (29/2)*Pi, (29/4)*Pi, (31/2)*Pi, (33/2)*Pi, (33/4)*Pi, (35/2)*Pi, (37/2)*Pi, (37/4)*Pi, (39/2)*Pi, (41/2)*Pi, (41/4)*Pi, (45/4)*Pi, (49/4)*Pi, (53/4)*Pi, (57/4)*Pi, (61/4)*Pi, (65/4)*Pi, (69/4)*Pi, (73/4)*Pi, (77/4)*Pi, (81/4)*Pi}

 

0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0, 0 = 0

(1)

 


 

Download manySols3.mw

for a given ODE system with fixed boundary conditions, it is "difficult" to imagine why a numerical solving process would give two different solutions - possible I suppose, but just very (very!) unlikely.

Thus you are going to have to demonstrate exactly how the same ODEs with the same initial/boundary conditions produces two different solutions. Upload the Matlab file which shows this phenomenon, if necessary

Meanwhile the attached worksheet shows the (unique) Maple solutions after I "strip out" all the random, unnecessary cr*p in your original worksheet

  restart:
  with(plots):

  bet:= 3.1: lam:= 0.01: pr:= 0.1: af:= 0.1:
   sf:= 0.1:  sc:= 0.1:   N:= 8:
  eq1:= diff(f(eta), eta$3)*(1+1/bet)+diff(f(eta), eta$2)*f(eta)-diff(f(eta), eta)^2+1+lam*t(eta) = 0:
  eq2:= diff(t(eta), eta$2)/pr+diff(t(eta), eta)*f(eta)-diff(f(eta), eta)*t(eta)+af*diff(c(eta), eta$2) = 0:
  eq3:= diff(t(eta), eta$2)/sc+diff(c(eta), eta)*f(eta)-diff(f(eta), eta)*c(eta)+sf*diff(t(eta), eta$2) = 0:
  bc:= f(0) = 0, D(f)(0) = ca, D(f)(N) = 1, t(0) = 1,
       t(N) = 0,    c(0) = 1,     c(N) = 0:

  caVals:= [-1.25, -1.24, -1.23]:
  colors:= [red, blue, green]:
  display
  ( [ seq
      ( odeplot
        ( dsolve
          ( eval
            ( [bc, eq1, eq2, eq3],
              ca = caVals[k]
            ),
            numeric,
            method = bvp[midrich]
          ),
          [eta, D(f)(eta)],
          color = colors[k]
        ),
        k = 1 .. numelems(caVals)
      )
    ]
  );

 

 

Download odeProb.mw

shown in the attached


 

  expr:=sin(9*x-(1/3)*Pi) = sin(7*x-(1/3)*Pi);
  ans:=simplify~([solve(expr,x)], symbolic);
#
# Sort in ascending order
#
  ans[sort(evalf~(%),output=permutation)];

-cos(9*x+(1/6)*Pi) = -cos(7*x+(1/6)*Pi)

 

[0, Pi, -(1/48)*Pi, (47/48)*Pi, (23/48)*Pi, -(25/48)*Pi, (11/48)*Pi, -(37/48)*Pi, -(13/48)*Pi, (35/48)*Pi, (5/48)*Pi, -(43/48)*Pi, -(19/48)*Pi, (29/48)*Pi, -(7/48)*Pi, (41/48)*Pi, (17/48)*Pi, -(31/48)*Pi]

 

[-(43/48)*Pi, -(37/48)*Pi, -(31/48)*Pi, -(25/48)*Pi, -(19/48)*Pi, -(13/48)*Pi, -(7/48)*Pi, -(1/48)*Pi, 0, (5/48)*Pi, (11/48)*Pi, (17/48)*Pi, (23/48)*Pi, (29/48)*Pi, (35/48)*Pi, (41/48)*Pi, (47/48)*Pi, Pi]

(1)

#
# A check
#
  [seq( `if`(eval(lhs(expr)-rhs(expr), x=-Pi/48+K*Pi/48)=0,-Pi/48+K*Pi/48,NULL), K=-46..49)];

[-(43/48)*Pi, -(37/48)*Pi, -(31/48)*Pi, -(25/48)*Pi, -(19/48)*Pi, -(13/48)*Pi, -(7/48)*Pi, -(1/48)*Pi, 0, (5/48)*Pi, (11/48)*Pi, (17/48)*Pi, (23/48)*Pi, (29/48)*Pi, (35/48)*Pi, (41/48)*Pi, (47/48)*Pi, Pi]

(2)

 


 

Download solveTrig.mw

 

some variant of the attached

restart;
expr1:=sin(x)*(x*y)^(1/3);
expr2:=sin(x)+(x*y)^(1/6);
getMatch:=expr-> `if`
                 ( ormap
                   ( patmatch,
                     [op(expr)],
                     `^`(a::name*b::name,c::nonunit(radnum)),
                     'la'
                   ),
                   printf("Pattern Match with c=%a\n", eval(c, la)),
                   printf("No Match\n")
                 ):
getMatch(expr1);
getMatch(expr2);

sin(x)*(x*y)^(1/3)

 

sin(x)+(x*y)^(1/6)

 

Pattern Match with c=1/3
Pattern Match with c=1/6

 

 

Download matchPat.mw

can be performed elementwise, as in the attached


 

#
# "Toy" example
#
  B:=Matrix( [ [ 1,   t,   t^2],
               [ 2, 2*t, 2*t^2],
               [ 3, 3*t, 3*t^2]
             ]
         );
#
# Elementwise (indefinite) integration
#
  int~(B, t);
#
# Elementwise (definite) integration
#
  int~(B, t=0..1);

Matrix(3, 3, {(1, 1) = 1, (1, 2) = t, (1, 3) = t^2, (2, 1) = 2, (2, 2) = 2*t, (2, 3) = 2*t^2, (3, 1) = 3, (3, 2) = 3*t, (3, 3) = 3*t^2})

 

Matrix(3, 3, {(1, 1) = t, (1, 2) = (1/2)*t^2, (1, 3) = (1/3)*t^3, (2, 1) = 2*t, (2, 2) = t^2, (2, 3) = (2/3)*t^3, (3, 1) = 3*t, (3, 2) = (3/2)*t^2, (3, 3) = t^3})

 

Matrix(%id = 18446744074573600158)

(1)

 

 

 


 

Download matInt.mw

when this kind of question has been asked before, the conclusion seems to be that the OP is using a "bootleg" copy of Maple. See for example

https://www.mapleprimes.com/questions/227464-Problem-With-Maple-2019
https://www.mapleprimes.com/questions/227376-Imaginary-To-Real

Obviously, if OP has a legitimate copy of Maple, then (s)he should contact MapleSoft support immediately with license details, etc

to first create a table whose entries are then used to initalise a vector when you can do the latter directly as in the attached


 

restart:

 M1 := 3;
 M2 := 3;
 nu := 1;
 for k1 from 0 while k1 <= M1-1 do
 for k2 from 0 while k2 <= M2-1 do
     GP[k1+1, k2+1] := simplify(sum((-1)^(k1-i1)*GAMMA(k1+i1+2*nu)*GAMMA(nu+1/2)*x^i1*(sum((-1)^(k2-i2)*GAMMA(k2+i2+2*nu)*GAMMA(nu+1/2)*y^i2/(GAMMA(i2+nu+1/2)*factorial(k2-i2)*factorial(i2)*GAMMA(2*nu)), i2 = 0 .. k2))/(GAMMA(i1+nu+1/2)*factorial(k1-i1)*factorial(i1)*GAMMA(2*nu)), i1 = 0 .. k1));
end do end do;

3

 

3

 

1

(1)

#
# Examine table entries (used for checking purposes only)
#
  GP();

(table( [( 3, 3 ) = (4*x-3)*(4*y-3)*(4*y-1)*(4*x-1), ( 1, 3 ) = 16*y^2-16*y+3, ( 3, 1 ) = 16*x^2-16*x+3, ( 2, 2 ) = (16*y-8)*x-8*y+4, ( 1, 2 ) = -2+4*y, ( 3, 2 ) = 2*(4*x-3)*(-1+2*y)*(4*x-1), ( 2, 3 ) = 2*(4*y-3)*(-1+2*x)*(4*y-1), ( 2, 1 ) = -2+4*x, ( 1, 1 ) = 1 ] ))()

(2)

#
# Instead of using loops to create a table, and using
# this to initializea vector, create the (column) vector
# directly
#
  restart;
  M1 := 3;
  M2 := 3;
  nu := 1;
  VGP:=<seq(seq( simplify(add((-1)^(k1-i1)*GAMMA(k1+i1+2*nu)*GAMMA(nu+1/2)*x^i1*(add((-1)^(k2-i2)*GAMMA(k2+i2+2*nu)*GAMMA(nu+1/2)*y^i2/(GAMMA(i2+nu+1/2)*factorial(k2-i2)*factorial(i2)*GAMMA(2*nu)), i2 = 0 .. k2))/(GAMMA(i1+nu+1/2)*factorial(k1-i1)*factorial(i1)*GAMMA(2*nu)), i1 = 0 .. k1)), k2=0..M2-1), k1=0..M1-1)>;

Vector(9, {(1) = 1, (2) = -2+4*y, (3) = 16*y^2-16*y+3, (4) = -2+4*x, (5) = (16*y-8)*x-8*y+4, (6) = (8*y-6)*(-1+2*x)*(4*y-1), (7) = 16*x^2-16*x+3, (8) = (8*x-6)*(-1+2*y)*(4*x-1), (9) = (4*x-3)*(4*y-3)*(4*y-1)*(4*x-1)})

(3)

#
# Use the same approach to create a vector of "parameters"
#
  A:=<seq( seq( a[k1,k2], k2=1..M2), k1=1..M1)>;

Vector(9, {(1) = a[1, 1], (2) = a[1, 2], (3) = a[1, 3], (4) = a[2, 1], (5) = a[2, 2], (6) = a[2, 3], (7) = a[3, 1], (8) = a[3, 2], (9) = a[3, 3]})

(4)

#
# Assuming that the 'function' defining the above parameters
# is 'known' - say k1^3 + k2, then one would have
#
  A:=<seq( seq( k1^3 + k2, k2=1..M2), k1=1..M1)>;

Vector(9, {(1) = 2, (2) = 3, (3) = 4, (4) = 9, (5) = 10, (6) = 11, (7) = 28, (8) = 29, (9) = 30})

(5)

 


 

Download initMat.mw

you could use the plottools:-translate() command, as in the attached

  with(plottools):
  with(plots):
  f:=t->piecewise( t<1 and t>=-1,t/2+1/2,
                   t>=1 and t<2,-t+2
                  );
  p1:=plot(f(t), t=-1..2);
  display([seq( translate(p1, 3*j, 0), j=-1..1)])

f := proc (t) options operator, arrow; piecewise(-1 <= t and t < 1, (1/2)*t+1/2, 1 <= t and t < 2, -t+2) end proc

 

 

 

 

Download perPlot.mw

Have you considered using the DynamicSystems() package?

The attached shows how to define a dynamic system based directly on your ODEs which can then be used to produce the "state-space" representation.

In the attached, I have assumed that v1(t), v2(t), v3(t) are input variables; i1(t), i2(t), and i3(t) are state variables; diff(i1(t),t),  diff(i2(t),t) and diff(i3(t),t) are outputs

  with(DynamicSystems):
  eq1:=(L1/n12 + n12*L2)*diff(i2(t), t) + L1*diff(i3(t), t)/n13 = -v1(t) - R1*i3(t)/n13 - R1*i2(t)/n12 + n12*v2(t) - n12*R2*i2(t):
  eq2:=L1*diff(i2(t), t)/n12 + (L1/n13 + n13*L3)*diff(i3(t), t) = -v1(t) - R1*i3(t)/n13 - R1*i2(t)/n12 + n13*v3(t) - n13*R3*i3(t):
  eq3:=-L2*diff(i2(t), t) + L1*diff(i1(t), t)/n12 = -v2(t) + R2*i2(t) + v1(t)/n12 - R1*i1(t)/n12:
#
# Define a dynamic system based on ODES
#
  sys:=DiffEquation( [ eq1,
                       eq2,
                       eq3,
                       y1(t)=diff(i1(t),t),
                       y2(t)=diff(i2(t),t),
                       y3(t)=diff(i3(t),t)
                     ],
                     [ v1(t), v2(t), v3(t)],
                     [ y1(t), y2(t), y3(t)],
                     statevariable=[i1(t), i2(t), i3(t)]
                   ):
#
# Examine the ODE system definition
#
  PrintSystem(sys);
#
# Produce the State-Space version of this system
#
  sys2:=StateSpace(sys):
#
# Examine the state-space system definition
# Will display the "standard" A, B, C, D
# state=-space matrices
#
  PrintSystem(sys2);

"[[[`Diff. Equation`],[continuous],[`3 output(s); 3 input(s)`],[inputvariable=[v1(t),v2(t),v3(t)]],[outputvariable=[y1(t),y2(t),y3(t)]],[de={[[[(L1/n12+n12 L2) ((&DifferentialD;)/(&DifferentialD;t) i2(t))+(L1 ((&DifferentialD;)/(&DifferentialD;t) i3(t)))/n13=-v1(t)-(R1 i3(t))/n13-(R1 i2(t))/n12+n12 v2(t)-n12 R2 i2(t),],[  (L1 ((&DifferentialD;)/(&DifferentialD;t) i2(t)))/n12+(L1/n13+n13 L3) ((&DifferentialD;)/(&DifferentialD;t) i3(t))=-v1(t)-(R1 i3(t))/n13-(R1 i2(t))/n12+n13 v3(t)-n13 R3 i3(t),],[  -L2 ((&DifferentialD;)/(&DifferentialD;t) i2(t))+(L1 ((&DifferentialD;)/(&DifferentialD;t) i1(t)))/n12=-v2(t)+R2 i2(t)+(v1(t))/n12-(R1 i1(t))/n12,],[  y1(t)=(&DifferentialD;)/(&DifferentialD;t) i1(t),],[  y2(t)=(&DifferentialD;)/(&DifferentialD;t) i2(t),],[  y3(t)=(&DifferentialD;)/(&DifferentialD;t) i3(t)]]]]]"

 

"[[[`State Space`],[continuous],[`3 output(s); 3 input(s); 3 state(s)`],[inputvariable=[v1(t),v2(t),v3(t)]],[outputvariable=[y1(t),y2(t),y3(t)]],[statevariable=[i1(t),i2(t),i3(t)]],[a=[[[-(L2 L3 R1 n12^2 n13^2+L1 L2 R1 n12^2+L1 L3 R1 n13^2)/(L1 (L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2)),-(-L1 L3 R2 n12 n13^2+L2 L3 R1 n12 n13^2)/(L1 (L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2)),-(-L1 L2 R3 n12^2 n13+L2 L3 R1 n12^2 n13)/(L1 (L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2))],[0,(-L3 R2 n12^2 n13^2-L1 R2 n12^2-L3 R1 n13^2)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2),(L1 R3 n12 n13-L3 R1 n12 n13)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2)],[0,-(-L1 R2 n12 n13+L2 R1 n12 n13)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2),-(L2 R3 n12^2 n13^2+L1 R3 n13^2+L2 R1 n12^2)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2)]]]],[b=[[[-(-L2 n12^2 L1-L1 L3 n13^2)/(L1 (L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2)),-(L3 n12 n13^2)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2),-(L2 n12^2 n13)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2)],[-(L3 n12 n13^2)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2),(L3 n12^2 n13^2+L1 n12^2)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2),-(L1 n12 n13)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2)],[-(L2 n12^2 n13)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2),-(L1 n12 n13)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2),-(-L2 n12^2 n13^2-L1 n13^2)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2)]]]],[c=[[[-(L2 L3 R1 n12^2 n13^2+L1 L2 R1 n12^2+L1 L3 R1 n13^2)/(L1 (L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2)),-(-L1 L3 R2 n12 n13^2+L2 L3 R1 n12 n13^2)/(L1 (L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2)),-(-L1 L2 R3 n12^2 n13+L2 L3 R1 n12^2 n13)/(L1 (L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2))],[0,(-L3 R2 n12^2 n13^2-L1 R2 n12^2-L3 R1 n13^2)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2),(L1 R3 n12 n13-L3 R1 n12 n13)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2)],[0,-(-L1 R2 n12 n13+L2 R1 n12 n13)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2),-(L2 R3 n12^2 n13^2+L1 R3 n13^2+L2 R1 n12^2)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2)]]]],[d=[[[-(-L2 n12^2 L1-L1 L3 n13^2)/(L1 (L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2)),-(L3 n12 n13^2)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2),-(L2 n12^2 n13)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2)],[-(L3 n12 n13^2)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2),(L3 n12^2 n13^2+L1 n12^2)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2),-(L1 n12 n13)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2)],[-(L2 n12^2 n13)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2),-(L1 n12 n13)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2),-(-L2 n12^2 n13^2-L1 n13^2)/(L2 n13^2 n12^2 L3+L2 n12^2 L1+L1 L3 n13^2)]]]]]"

(1)

 

 


 

Download DynSys.mw

both of the "solutions" you provide appear to be (slightly) wrong. I suppose it is possible that the "solutions" are correct and the ODEs are wrong

Not really sure why you would want to use numerical methods for these systems when Maple will provide exact analtical solutions, but it can be done, and I suppose(?) may be useful for comparison purposes

The attached shows analytical and numerical solutions for both of your examples

  restart;

#############
# Example 1 #
#############
#
# Define ODE and boundary conditions
#
  ode:=diff(y(x),x,x)+10*y(x)=99*sin(x);
  bcs:=y(0)=1, D(y)(0)=11;
#
# Solve ODE analytically
#
  sol1A:=dsolve([ode, bcs]);
#
# Solve ODE numerically
#
  sol1B:=dsolve([ode, bcs], numeric);
#
# Plot the analytic and numeric solutions
# separately, and then on the on the same
# graph to see if there are any significant
# differences - there aren't!
#
  p1:=plot( rhs(sol1A),
            x=0..100,
            color=red,
            title="Analytic Solution",
            titlefont=[times, bold,20]
          );
  p2:=plots:-odeplot( sol1B,
                      [x, y(x)],
                      x=0..100,
                      color=blue,
                      title="Numerical Solution",
                      titlefont=[times, bold,20]
                    );
  plots:-display( [p1,p2],
                  title="Both Solutions",
                  titlefont=[times, bold,20]
                );

diff(diff(y(x), x), x)+10*y(x) = 99*sin(x)

 

y(0) = 1, (D(y))(0) = 11

 

y(x) = cos(10^(1/2)*x)+11*sin(x)

 

proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_rkf45) else _xout := evalf(x_rkf45) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _dtbl, _dat, _vmap, _x0, _y0, _val, _dig, _n, _ne, _nd, _nv, _pars, _ini, _par, _i, _j, _k, _src; option `Copyright (c) 2002 by Waterloo Maple Inc. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _pars := []; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 26, [( 1 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 2 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 3 ) = ([0, 0, 0, Array(1..0, 1..2, {}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..63, {(1) = 2, (2) = 2, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 1, (8) = 0, (9) = 0, (10) = 0, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 1, (19) = 30000, (20) = 0, (21) = 0, (22) = 1, (23) = 4, (24) = 0, (25) = 1, (26) = 15, (27) = 1, (28) = 0, (29) = 1, (30) = 3, (31) = 3, (32) = 0, (33) = 1, (34) = 0, (35) = 0, (36) = 0, (37) = 0, (38) = 0, (39) = 0, (40) = 0, (41) = 0, (42) = 0, (43) = 1, (44) = 0, (45) = 0, (46) = 0, (47) = 0, (48) = 0, (49) = 0, (50) = 50, (51) = 1, (52) = 0, (53) = 0, (54) = 0, (55) = 0, (56) = 0, (57) = 0, (58) = 0, (59) = 10000, (60) = 0, (61) = 1000, (62) = 0, (63) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = 0.5047658755841546e-2, (7) = .0, (8) = 0.10e-5, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = 1.0, (14) = .0, (15) = .49999999999999, (16) = .0, (17) = 1.0, (18) = 1.0, (19) = .0, (20) = .0, (21) = 1.0, (22) = 1.0, (23) = .0, (24) = .0, (25) = 0.10e-14, (26) = .0, (27) = .0, (28) = .0}, datatype = float[8], order = C_order)), ( 6 ) = (Array(1..2, {(1) = 1.0, (2) = 11.0}, datatype = float[8], order = C_order)), ( 7 ) = ([Array(1..4, 1..7, {(1, 1) = .0, (1, 2) = .203125, (1, 3) = .3046875, (1, 4) = .75, (1, 5) = .8125, (1, 6) = .40625, (1, 7) = .8125, (2, 1) = 0.6378173828125e-1, (2, 2) = .0, (2, 3) = .279296875, (2, 4) = .27237892150878906, (2, 5) = -0.9686851501464844e-1, (2, 6) = 0.1956939697265625e-1, (2, 7) = .5381584167480469, (3, 1) = 0.31890869140625e-1, (3, 2) = .0, (3, 3) = -.34375, (3, 4) = -.335235595703125, (3, 5) = .2296142578125, (3, 6) = .41748046875, (3, 7) = 11.480712890625, (4, 1) = 0.9710520505905151e-1, (4, 2) = .0, (4, 3) = .40350341796875, (4, 4) = 0.20297467708587646e-1, (4, 5) = -0.6054282188415527e-2, (4, 6) = -0.4770040512084961e-1, (4, 7) = .77858567237854}, datatype = float[8], order = C_order), Array(1..6, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = 1.0, (2, 1) = .25, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = 1.0, (3, 1) = .1875, (3, 2) = .5625, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = 2.0, (4, 1) = .23583984375, (4, 2) = -.87890625, (4, 3) = .890625, (4, 4) = .0, (4, 5) = .0, (4, 6) = .2681884765625, (5, 1) = .1272735595703125, (5, 2) = -.5009765625, (5, 3) = .44921875, (5, 4) = -0.128936767578125e-1, (5, 5) = .0, (5, 6) = 0.626220703125e-1, (6, 1) = -0.927734375e-1, (6, 2) = .626220703125, (6, 3) = -.4326171875, (6, 4) = .1418304443359375, (6, 5) = -0.861053466796875e-1, (6, 6) = .3131103515625}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .386, (3) = .21, (4) = .63, (5) = 1.0, (6) = 1.0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .25, (2) = -.1043, (3) = .1035, (4) = -0.362e-1, (5) = .0, (6) = .0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 1.544, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = .9466785280815533, (3, 2) = .25570116989825814, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = 3.3148251870684886, (4, 2) = 2.896124015972123, (4, 3) = .9986419139977808, (4, 4) = .0, (4, 5) = .0, (5, 1) = 1.2212245092262748, (5, 2) = 6.019134481287752, (5, 3) = 12.537083329320874, (5, 4) = -.687886036105895, (5, 5) = .0, (6, 1) = 1.2212245092262748, (6, 2) = 6.019134481287752, (6, 3) = 12.537083329320874, (6, 4) = -.687886036105895, (6, 5) = 1.0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = -5.6688, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = -2.4300933568337584, (3, 2) = -.20635991570891224, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = -.10735290581452621, (4, 2) = -9.594562251021896, (4, 3) = -20.470286148096154, (4, 4) = .0, (4, 5) = .0, (5, 1) = 7.496443313968615, (5, 2) = -10.246804314641219, (5, 3) = -33.99990352819906, (5, 4) = 11.708908932061595, (5, 5) = .0, (6, 1) = 8.083246795922411, (6, 2) = -7.981132988062785, (6, 3) = -31.52159432874373, (6, 4) = 16.319305431231363, (6, 5) = -6.0588182388340535}, datatype = float[8], order = C_order), Array(1..3, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 10.126235083446911, (2, 2) = -7.487995877607633, (2, 3) = -34.800918615557414, (2, 4) = -7.9927717075687275, (2, 5) = 1.0251377232956207, (3, 1) = -.6762803392806898, (3, 2) = 6.087714651678606, (3, 3) = 16.43084320892463, (3, 4) = 24.767225114183653, (3, 5) = -6.5943891257167815}, datatype = float[8], order = C_order)]), ( 9 ) = ([Array(1..2, {(1) = .1, (2) = .1}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 0, (2) = 0}, datatype = integer[8]), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 0, (2) = 0}, datatype = integer[8])]), ( 8 ) = ([Array(1..2, {(1) = 1.0, (2) = 11.0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 11.0, (2) = -10.0}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..2, {(1, 1) = .0, (1, 2) = .0, (2, 0) = .0, (2, 1) = .0, (2, 2) = .0, (3, 0) = .0, (3, 1) = .0, (3, 2) = .0, (4, 0) = .0, (4, 1) = .0, (4, 2) = .0, (5, 0) = .0, (5, 1) = .0, (5, 2) = .0, (6, 0) = .0, (6, 1) = .0, (6, 2) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = y(x), Y[2] = diff(y(x),x)]`; YP[2] := 99*sin(X)-10*Y[1]; YP[1] := Y[2]; 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = y(x), Y[2] = diff(y(x),x)]`; YP[2] := 99*sin(X)-10*Y[1]; YP[1] := Y[2]; 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0)  ] ))  ] ); _y0 := Array(0..2, {(1) = 0., (2) = 1.}); _vmap := array( 1 .. 2, [( 1 ) = (1), ( 2 ) = (2)  ] ); _x0 := _dtbl[1][5][5]; _n := _dtbl[1][4][1]; _ne := _dtbl[1][4][3]; _nd := _dtbl[1][4][4]; _nv := _dtbl[1][4][16]; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then if _Env_smart_dsolve_numeric = true or _dtbl[1][4][10] = 1 then if _xout = "left" then if type(_dtbl[2], 'table') then return _dtbl[2][5][1] end if elif _xout = "right" then if type(_dtbl[3], 'table') then return _dtbl[3][5][1] end if end if end if; return _dtbl[1][5][5] elif _xout = "method" then return _dtbl[1][15] elif _xout = "storage" then return evalb(_dtbl[1][4][10] = 1) elif _xout = "leftdata" then if not type(_dtbl[2], 'array') then return NULL else return eval(_dtbl[2]) end if elif _xout = "rightdata" then if not type(_dtbl[3], 'array') then return NULL else return eval(_dtbl[3]) end if elif _xout = "enginedata" then return eval(_dtbl[1]) elif _xout = "enginereset" then _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); return NULL elif _xout = "initial" then return procname(_y0[0]) elif _xout = "laxtol" then return _dtbl[`if`(member(_dtbl[4], {2, 3}), _dtbl[4], 1)][5][18] elif _xout = "numfun" then return `if`(member(_dtbl[4], {2, 3}), _dtbl[_dtbl[4]][4][18], 0) elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return procname(_y0[0]), [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _dtbl[4] <> 2 and _dtbl[4] <> 3 or _x0-_dtbl[_dtbl[4]][5][1] = 0. then error "no information is available on last computed point" else _xout := _dtbl[_dtbl[4]][5][1] end if elif _xout = "function" then if _dtbl[1][4][33]-2. = 0 then return eval(_dtbl[1][10], 1) else return eval(_dtbl[1][10][1], 1) end if elif _xout = "map" then return copy(_vmap) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); _i := false; if _par <> [] then _i := `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then _i := `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) or _i end if; if _i then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars); if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') and _dtbl[1][4][10] <> 1 then procname("right") := _y0[0]; procname("left") := _y0[0] end if end if; if _xout = "initial" then return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)] elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] else return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)], [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] end if elif _xin = "eventstop" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then return 0 end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 <= _dtbl[5-_i][4][9] then _i := 5-_i; _dtbl[4] := _i; _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) elif 100 <= _dtbl[_i][4][9] then _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) else return 0 end if elif _xin = "eventstatus" then if _nv = 0 then error "this solution has no events" end if; _i := [selectremove(proc (a) options operator, arrow; _dtbl[1][3][1][a, 7] = 1 end proc, {seq(_j, _j = 1 .. round(_dtbl[1][3][1][_nv+1, 1]))})]; return ':-enabled' = _i[1], ':-disabled' = _i[2] elif _xin = "eventclear" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then error "no events to clear" end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 < _dtbl[5-_i][4][9] then _dtbl[4] := 5-_i; _i := 5-_i end if; if _dtbl[_i][4][9] < 100 then error "no events to clear" elif _nv < _dtbl[_i][4][9]-100 then error "event error condition cannot be cleared" else _j := _dtbl[_i][4][9]-100; if irem(round(_dtbl[_i][3][1][_j, 4]), 2) = 1 then error "retriggerable events cannot be cleared" end if; _j := round(_dtbl[_i][3][1][_j, 1]); for _k to _nv do if _dtbl[_i][3][1][_k, 1] = _j then if _dtbl[_i][3][1][_k, 2] = 3 then error "range events cannot be cleared" end if; _dtbl[_i][3][1][_k, 8] := _dtbl[_i][3][1][_nv+1, 8] end if end do; _dtbl[_i][4][17] := 0; _dtbl[_i][4][9] := 0; if _dtbl[1][4][10] = 1 then if _i = 2 then try procname(procname("left")) catch:  end try else try procname(procname("right")) catch:  end try end if end if end if; return  elif type(_xin, `=`) and member(lhs(_xin), {"eventdisable", "eventenable"}) then if _nv = 0 then error "this solution has no events" end if; if type(rhs(_xin), {('list')('posint'), ('set')('posint')}) then _i := {op(rhs(_xin))} elif type(rhs(_xin), 'posint') then _i := {rhs(_xin)} else error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; if select(proc (a) options operator, arrow; _nv < a end proc, _i) <> {} then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _k := {}; for _j to _nv do if member(round(_dtbl[1][3][1][_j, 1]), _i) then _k := `union`(_k, {_j}) end if end do; _i := _k; if lhs(_xin) = "eventdisable" then _dtbl[4] := 0; _j := [evalb(assigned(_dtbl[2]) and member(_dtbl[2][4][17], _i)), evalb(assigned(_dtbl[3]) and member(_dtbl[3][4][17], _i))]; for _k in _i do _dtbl[1][3][1][_k, 7] := 0; if assigned(_dtbl[2]) then _dtbl[2][3][1][_k, 7] := 0 end if; if assigned(_dtbl[3]) then _dtbl[3][3][1][_k, 7] := 0 end if end do; if _j[1] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[2][3][4][_k, 1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to defined init `, _dtbl[2][3][4][_k, 1]); _dtbl[2][3][1][_k, 8] := _dtbl[2][3][4][_k, 1] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to rate hysteresis init `, _dtbl[2][5][24]); _dtbl[2][3][1][_k, 8] := _dtbl[2][5][24] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to initial init `, _x0); _dtbl[2][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to fireinitial init `, _x0-1); _dtbl[2][3][1][_k, 8] := _x0-1 end if end do; _dtbl[2][4][17] := 0; _dtbl[2][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("left")) end if end if; if _j[2] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[3][3][4][_k, 2], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to defined init `, _dtbl[3][3][4][_k, 2]); _dtbl[3][3][1][_k, 8] := _dtbl[3][3][4][_k, 2] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to rate hysteresis init `, _dtbl[3][5][24]); _dtbl[3][3][1][_k, 8] := _dtbl[3][5][24] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to initial init `, _x0); _dtbl[3][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to fireinitial init `, _x0+1); _dtbl[3][3][1][_k, 8] := _x0+1 end if end do; _dtbl[3][4][17] := 0; _dtbl[3][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("right")) end if end if else for _k in _i do _dtbl[1][3][1][_k, 7] := 1 end do; _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); _dtbl[4] := 0; if _dtbl[1][4][10] = 1 then if _x0 <= procname("right") then try procname(procname("right")) catch:  end try end if; if procname("left") <= _x0 then try procname(procname("left")) catch:  end try end if end if end if; return  elif type(_xin, `=`) and lhs(_xin) = "eventfired" then if not type(rhs(_xin), 'list') then error "'eventfired' must be specified as a list" end if; if _nv = 0 then error "this solution has no events" end if; if _dtbl[4] <> 2 and _dtbl[4] <> 3 then error "'direction' must be set prior to calling/setting 'eventfired'" end if; _i := _dtbl[4]; _val := NULL; if not assigned(_EnvEventRetriggerWarned) then _EnvEventRetriggerWarned := false end if; for _k in rhs(_xin) do if type(_k, 'integer') then _src := _k elif type(_k, 'integer' = 'anything') and type(evalf(rhs(_k)), 'numeric') then _k := lhs(_k) = evalf[max(Digits, 18)](rhs(_k)); _src := lhs(_k) else error "'eventfired' entry is not valid: %1", _k end if; if _src < 1 or round(_dtbl[1][3][1][_nv+1, 1]) < _src then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _src := {seq(`if`(_dtbl[1][3][1][_j, 1]-_src = 0., _j, NULL), _j = 1 .. _nv)}; if nops(_src) <> 1 then error "'eventfired' can only be set/queried for root-finding events and time/interval events" end if; _src := _src[1]; if _dtbl[1][3][1][_src, 2] <> 0. and _dtbl[1][3][1][_src, 2]-2. <> 0. then error "'eventfired' can only be set/queried for root-finding events and time/interval events" elif irem(round(_dtbl[1][3][1][_src, 4]), 2) = 1 then if _EnvEventRetriggerWarned = false then WARNING(`'eventfired' has no effect on events that retrigger`) end if; _EnvEventRetriggerWarned := true end if; if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then _val := _val, undefined elif type(_dtbl[_i][3][4][_src, _i-1], 'undefined') or _i = 2 and _dtbl[2][3][1][_src, 8] < _dtbl[2][3][4][_src, 1] or _i = 3 and _dtbl[3][3][4][_src, 2] < _dtbl[3][3][1][_src, 8] then _val := _val, _dtbl[_i][3][1][_src, 8] else _val := _val, _dtbl[_i][3][4][_src, _i-1] end if; if type(_k, `=`) then if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then error "cannot set event code for a rate hysteresis event" end if; userinfo(3, {'events', 'eventreset'}, `manual set event code `, _src, ` to value `, rhs(_k)); _dtbl[_i][3][1][_src, 8] := rhs(_k); _dtbl[_i][3][4][_src, _i-1] := rhs(_k) end if end do; return [_val] elif type(_xin, `=`) and lhs(_xin) = "direction" then if not member(rhs(_xin), {-1, 1, ':-left', ':-right'}) then error "'direction' must be specified as either '1' or 'right' (positive) or '-1' or 'left' (negative)" end if; _src := `if`(_dtbl[4] = 2, -1, `if`(_dtbl[4] = 3, 1, undefined)); _i := `if`(member(rhs(_xin), {1, ':-right'}), 3, 2); _dtbl[4] := _i; _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if; return _src elif _xin = "eventcount" then if _dtbl[1][3][1] = 0 or _dtbl[4] <> 2 and _dtbl[4] <> 3 then return 0 else return round(_dtbl[_dtbl[4]][3][1][_nv+1, 12]) end if else return "procname" end if end if; if _xout = _x0 then return [_x0, seq(evalf(_dtbl[1][6][_vmap[_i]]), _i = 1 .. _n-_ne)] end if; _i := `if`(_x0 <= _xout, 3, 2); if _xin = "last" and 0 < _dtbl[_i][4][9] and _dtbl[_i][4][9] < 100 then _dat := eval(_dtbl[_i], 2); _j := _dat[4][20]; return [_dat[11][_j, 0], seq(_dat[11][_j, _vmap[_i]], _i = 1 .. _n-_ne-_nd), seq(_dat[8][1][_vmap[_i]], _i = _n-_ne-_nd+1 .. _n-_ne)] end if; if not type(_dtbl[_i], 'array') then _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if end if; if _xin <> "last" then if 0 < 0 then if `dsolve/numeric/checkglobals`(op(_dtbl[1][14]), _pars, _n, _y0) then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars, _i) end if end if; if _dtbl[1][4][7] = 0 then error "parameters must be initialized before solution can be computed" end if end if; _dat := eval(_dtbl[_i], 2); _dtbl[4] := _i; try _src := `dsolve/numeric/SC/IVPrun`(_dat, _xout) catch: userinfo(2, `dsolve/debug`, print(`Exception in solnproc:`, [lastexception][2 .. -1])); error  end try; if _dat[17] <> _dtbl[1][17] then _dtbl[1][17] := _dat[17]; _dtbl[1][10] := _dat[10] end if; if _src = 0 and 100 < _dat[4][9] then _val := _dat[3][1][_nv+1, 8] else _val := _dat[11][_dat[4][20], 0] end if; if _src <> 0 or _dat[4][9] <= 0 then _dtbl[1][5][1] := _xout else _dtbl[1][5][1] := _val end if; if _i = 3 and _val < _xout then Rounding := -infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further right of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further right of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further right of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further right of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further right of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further right of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further right of %1", evalf[8](_val) end if elif _i = 2 and _xout < _val then Rounding := infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further left of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further left of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further left of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further left of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further left of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further left of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further left of %1", evalf[8](_val) end if end if; if _EnvInFsolve = true then _dig := _dat[4][26]; if type(_EnvDSNumericSaveDigits, 'posint') then _dat[4][26] := _EnvDSNumericSaveDigits else _dat[4][26] := Digits end if; _Env_dsolve_SC_native := true; if _dat[4][25] = 1 then _i := 1; _dat[4][25] := 2 else _i := _dat[4][25] end if; _val := `dsolve/numeric/SC/IVPval`(_dat, _xout, _src); _dat[4][25] := _i; _dat[4][26] := _dig; [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] else Digits := _dat[4][26]; _val := `dsolve/numeric/SC/IVPval`(eval(_dat, 2), _xout, _src); [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] end if end proc, (2) = Array(0..0, {}), (3) = [x, y(x), diff(y(x), x)], (4) = []}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_rkf45, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_rkf45, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_rkf45, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_rkf45, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_rkf45), 'string') = rhs(x_rkf45); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_rkf45), 'string') = rhs(x_rkf45)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_rkf45) else _ndsol := 1; _ndsol := _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_rkf45) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error  end try end proc

 

 

 

 

#############
# Example 2 #
#############
#
# Define ODE system
#
  ode:= diff(y(x),x)=z(x), diff(z(x),x)=-y(x)+x;
  bcs:=y(0)=1, z(0)=2;
#
# Solve ODE system analytically
#
  sol2A:=dsolve( [ode, bcs]);
#
# Solve ODE system numerically
#
  sol2B:=dsolve( [ode, bcs], numeric);
#
# Plot the analytic and numeric solutions
# separately, and then on the on the same
# graph to see if there are any significant
# differences - there aren't!
#
  p3:=plot( [rhs(sol2A[1]), rhs(sol2A[2])],
            x=0..100,
            color=[red, blue],
            title="Analytic Solution",
            titlefont=[times, bold,20]
          );
  p4:= plots:-odeplot( sol2B,
                       [[x, y(x)], [x, z(x)]],
                       x=0..100,
                       color=[black,green],
                       title="Numerical Solution",
                       titlefont=[times, bold,20]
                     );
  plots:-display([p3, p4],
                  title="Both Solutions",
                  titlefont=[times, bold,20]);

diff(y(x), x) = z(x), diff(z(x), x) = -y(x)+x

 

y(0) = 1, z(0) = 2

 

{y(x) = sin(x)+cos(x)+x, z(x) = cos(x)-sin(x)+1}

 

proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_rkf45) else _xout := evalf(x_rkf45) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _dtbl, _dat, _vmap, _x0, _y0, _val, _dig, _n, _ne, _nd, _nv, _pars, _ini, _par, _i, _j, _k, _src; option `Copyright (c) 2002 by Waterloo Maple Inc. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _pars := []; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 26, [( 1 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 2 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 3 ) = ([0, 0, 0, Array(1..0, 1..2, {}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..63, {(1) = 2, (2) = 2, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 1, (8) = 0, (9) = 0, (10) = 0, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 1, (19) = 30000, (20) = 0, (21) = 0, (22) = 1, (23) = 4, (24) = 0, (25) = 1, (26) = 15, (27) = 1, (28) = 0, (29) = 1, (30) = 3, (31) = 3, (32) = 0, (33) = 1, (34) = 0, (35) = 0, (36) = 0, (37) = 0, (38) = 0, (39) = 0, (40) = 0, (41) = 0, (42) = 0, (43) = 1, (44) = 0, (45) = 0, (46) = 0, (47) = 0, (48) = 0, (49) = 0, (50) = 50, (51) = 1, (52) = 0, (53) = 0, (54) = 0, (55) = 0, (56) = 0, (57) = 0, (58) = 0, (59) = 10000, (60) = 0, (61) = 1000, (62) = 0, (63) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = 0.27762123157128504e-1, (7) = .0, (8) = 0.10e-5, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = 1.0, (14) = .0, (15) = .49999999999999, (16) = .0, (17) = 1.0, (18) = 1.0, (19) = .0, (20) = .0, (21) = 1.0, (22) = 1.0, (23) = .0, (24) = .0, (25) = 0.10e-14, (26) = .0, (27) = .0, (28) = .0}, datatype = float[8], order = C_order)), ( 6 ) = (Array(1..2, {(1) = 1.0, (2) = 2.0}, datatype = float[8], order = C_order)), ( 7 ) = ([Array(1..4, 1..7, {(1, 1) = .0, (1, 2) = .203125, (1, 3) = .3046875, (1, 4) = .75, (1, 5) = .8125, (1, 6) = .40625, (1, 7) = .8125, (2, 1) = 0.6378173828125e-1, (2, 2) = .0, (2, 3) = .279296875, (2, 4) = .27237892150878906, (2, 5) = -0.9686851501464844e-1, (2, 6) = 0.1956939697265625e-1, (2, 7) = .5381584167480469, (3, 1) = 0.31890869140625e-1, (3, 2) = .0, (3, 3) = -.34375, (3, 4) = -.335235595703125, (3, 5) = .2296142578125, (3, 6) = .41748046875, (3, 7) = 11.480712890625, (4, 1) = 0.9710520505905151e-1, (4, 2) = .0, (4, 3) = .40350341796875, (4, 4) = 0.20297467708587646e-1, (4, 5) = -0.6054282188415527e-2, (4, 6) = -0.4770040512084961e-1, (4, 7) = .77858567237854}, datatype = float[8], order = C_order), Array(1..6, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = 1.0, (2, 1) = .25, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = 1.0, (3, 1) = .1875, (3, 2) = .5625, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = 2.0, (4, 1) = .23583984375, (4, 2) = -.87890625, (4, 3) = .890625, (4, 4) = .0, (4, 5) = .0, (4, 6) = .2681884765625, (5, 1) = .1272735595703125, (5, 2) = -.5009765625, (5, 3) = .44921875, (5, 4) = -0.128936767578125e-1, (5, 5) = .0, (5, 6) = 0.626220703125e-1, (6, 1) = -0.927734375e-1, (6, 2) = .626220703125, (6, 3) = -.4326171875, (6, 4) = .1418304443359375, (6, 5) = -0.861053466796875e-1, (6, 6) = .3131103515625}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .386, (3) = .21, (4) = .63, (5) = 1.0, (6) = 1.0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .25, (2) = -.1043, (3) = .1035, (4) = -0.362e-1, (5) = .0, (6) = .0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 1.544, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = .9466785280815533, (3, 2) = .25570116989825814, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = 3.3148251870684886, (4, 2) = 2.896124015972123, (4, 3) = .9986419139977808, (4, 4) = .0, (4, 5) = .0, (5, 1) = 1.2212245092262748, (5, 2) = 6.019134481287752, (5, 3) = 12.537083329320874, (5, 4) = -.687886036105895, (5, 5) = .0, (6, 1) = 1.2212245092262748, (6, 2) = 6.019134481287752, (6, 3) = 12.537083329320874, (6, 4) = -.687886036105895, (6, 5) = 1.0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = -5.6688, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = -2.4300933568337584, (3, 2) = -.20635991570891224, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = -.10735290581452621, (4, 2) = -9.594562251021896, (4, 3) = -20.470286148096154, (4, 4) = .0, (4, 5) = .0, (5, 1) = 7.496443313968615, (5, 2) = -10.246804314641219, (5, 3) = -33.99990352819906, (5, 4) = 11.708908932061595, (5, 5) = .0, (6, 1) = 8.083246795922411, (6, 2) = -7.981132988062785, (6, 3) = -31.52159432874373, (6, 4) = 16.319305431231363, (6, 5) = -6.0588182388340535}, datatype = float[8], order = C_order), Array(1..3, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 10.126235083446911, (2, 2) = -7.487995877607633, (2, 3) = -34.800918615557414, (2, 4) = -7.9927717075687275, (2, 5) = 1.0251377232956207, (3, 1) = -.6762803392806898, (3, 2) = 6.087714651678606, (3, 3) = 16.43084320892463, (3, 4) = 24.767225114183653, (3, 5) = -6.5943891257167815}, datatype = float[8], order = C_order)]), ( 9 ) = ([Array(1..2, {(1) = .1, (2) = .1}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..2, {(1, 1) = .0, (1, 2) = .0, (2, 1) = .0, (2, 2) = .0}, datatype = float[8], order = C_order), Array(1..2, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 0, (2) = 0}, datatype = integer[8]), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..4, {(1) = .0, (2) = .0, (3) = .0, (4) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 0, (2) = 0}, datatype = integer[8])]), ( 8 ) = ([Array(1..2, {(1) = 1.0, (2) = 2.0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = 2.0, (2) = -1.0}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..2, {(1, 1) = .0, (1, 2) = .0, (2, 0) = .0, (2, 1) = .0, (2, 2) = .0, (3, 0) = .0, (3, 1) = .0, (3, 2) = .0, (4, 0) = .0, (4, 1) = .0, (4, 2) = .0, (5, 0) = .0, (5, 1) = .0, (5, 2) = .0, (6, 0) = .0, (6, 1) = .0, (6, 2) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = y(x), Y[2] = z(x)]`; YP[2] := -Y[1]+X; YP[1] := Y[2]; 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = y(x), Y[2] = z(x)]`; YP[2] := -Y[1]+X; YP[1] := Y[2]; 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0)  ] ))  ] ); _y0 := Array(0..2, {(1) = 0., (2) = 1.}); _vmap := array( 1 .. 2, [( 1 ) = (1), ( 2 ) = (2)  ] ); _x0 := _dtbl[1][5][5]; _n := _dtbl[1][4][1]; _ne := _dtbl[1][4][3]; _nd := _dtbl[1][4][4]; _nv := _dtbl[1][4][16]; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then if _Env_smart_dsolve_numeric = true or _dtbl[1][4][10] = 1 then if _xout = "left" then if type(_dtbl[2], 'table') then return _dtbl[2][5][1] end if elif _xout = "right" then if type(_dtbl[3], 'table') then return _dtbl[3][5][1] end if end if end if; return _dtbl[1][5][5] elif _xout = "method" then return _dtbl[1][15] elif _xout = "storage" then return evalb(_dtbl[1][4][10] = 1) elif _xout = "leftdata" then if not type(_dtbl[2], 'array') then return NULL else return eval(_dtbl[2]) end if elif _xout = "rightdata" then if not type(_dtbl[3], 'array') then return NULL else return eval(_dtbl[3]) end if elif _xout = "enginedata" then return eval(_dtbl[1]) elif _xout = "enginereset" then _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); return NULL elif _xout = "initial" then return procname(_y0[0]) elif _xout = "laxtol" then return _dtbl[`if`(member(_dtbl[4], {2, 3}), _dtbl[4], 1)][5][18] elif _xout = "numfun" then return `if`(member(_dtbl[4], {2, 3}), _dtbl[_dtbl[4]][4][18], 0) elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return procname(_y0[0]), [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _dtbl[4] <> 2 and _dtbl[4] <> 3 or _x0-_dtbl[_dtbl[4]][5][1] = 0. then error "no information is available on last computed point" else _xout := _dtbl[_dtbl[4]][5][1] end if elif _xout = "function" then if _dtbl[1][4][33]-2. = 0 then return eval(_dtbl[1][10], 1) else return eval(_dtbl[1][10][1], 1) end if elif _xout = "map" then return copy(_vmap) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); _i := false; if _par <> [] then _i := `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then _i := `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) or _i end if; if _i then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars); if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') and _dtbl[1][4][10] <> 1 then procname("right") := _y0[0]; procname("left") := _y0[0] end if end if; if _xout = "initial" then return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)] elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] else return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)], [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] end if elif _xin = "eventstop" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then return 0 end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 <= _dtbl[5-_i][4][9] then _i := 5-_i; _dtbl[4] := _i; _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) elif 100 <= _dtbl[_i][4][9] then _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) else return 0 end if elif _xin = "eventstatus" then if _nv = 0 then error "this solution has no events" end if; _i := [selectremove(proc (a) options operator, arrow; _dtbl[1][3][1][a, 7] = 1 end proc, {seq(_j, _j = 1 .. round(_dtbl[1][3][1][_nv+1, 1]))})]; return ':-enabled' = _i[1], ':-disabled' = _i[2] elif _xin = "eventclear" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then error "no events to clear" end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 < _dtbl[5-_i][4][9] then _dtbl[4] := 5-_i; _i := 5-_i end if; if _dtbl[_i][4][9] < 100 then error "no events to clear" elif _nv < _dtbl[_i][4][9]-100 then error "event error condition cannot be cleared" else _j := _dtbl[_i][4][9]-100; if irem(round(_dtbl[_i][3][1][_j, 4]), 2) = 1 then error "retriggerable events cannot be cleared" end if; _j := round(_dtbl[_i][3][1][_j, 1]); for _k to _nv do if _dtbl[_i][3][1][_k, 1] = _j then if _dtbl[_i][3][1][_k, 2] = 3 then error "range events cannot be cleared" end if; _dtbl[_i][3][1][_k, 8] := _dtbl[_i][3][1][_nv+1, 8] end if end do; _dtbl[_i][4][17] := 0; _dtbl[_i][4][9] := 0; if _dtbl[1][4][10] = 1 then if _i = 2 then try procname(procname("left")) catch:  end try else try procname(procname("right")) catch:  end try end if end if end if; return  elif type(_xin, `=`) and member(lhs(_xin), {"eventdisable", "eventenable"}) then if _nv = 0 then error "this solution has no events" end if; if type(rhs(_xin), {('list')('posint'), ('set')('posint')}) then _i := {op(rhs(_xin))} elif type(rhs(_xin), 'posint') then _i := {rhs(_xin)} else error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; if select(proc (a) options operator, arrow; _nv < a end proc, _i) <> {} then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _k := {}; for _j to _nv do if member(round(_dtbl[1][3][1][_j, 1]), _i) then _k := `union`(_k, {_j}) end if end do; _i := _k; if lhs(_xin) = "eventdisable" then _dtbl[4] := 0; _j := [evalb(assigned(_dtbl[2]) and member(_dtbl[2][4][17], _i)), evalb(assigned(_dtbl[3]) and member(_dtbl[3][4][17], _i))]; for _k in _i do _dtbl[1][3][1][_k, 7] := 0; if assigned(_dtbl[2]) then _dtbl[2][3][1][_k, 7] := 0 end if; if assigned(_dtbl[3]) then _dtbl[3][3][1][_k, 7] := 0 end if end do; if _j[1] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[2][3][4][_k, 1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to defined init `, _dtbl[2][3][4][_k, 1]); _dtbl[2][3][1][_k, 8] := _dtbl[2][3][4][_k, 1] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to rate hysteresis init `, _dtbl[2][5][24]); _dtbl[2][3][1][_k, 8] := _dtbl[2][5][24] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to initial init `, _x0); _dtbl[2][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to fireinitial init `, _x0-1); _dtbl[2][3][1][_k, 8] := _x0-1 end if end do; _dtbl[2][4][17] := 0; _dtbl[2][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("left")) end if end if; if _j[2] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[3][3][4][_k, 2], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to defined init `, _dtbl[3][3][4][_k, 2]); _dtbl[3][3][1][_k, 8] := _dtbl[3][3][4][_k, 2] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to rate hysteresis init `, _dtbl[3][5][24]); _dtbl[3][3][1][_k, 8] := _dtbl[3][5][24] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to initial init `, _x0); _dtbl[3][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to fireinitial init `, _x0+1); _dtbl[3][3][1][_k, 8] := _x0+1 end if end do; _dtbl[3][4][17] := 0; _dtbl[3][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("right")) end if end if else for _k in _i do _dtbl[1][3][1][_k, 7] := 1 end do; _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); _dtbl[4] := 0; if _dtbl[1][4][10] = 1 then if _x0 <= procname("right") then try procname(procname("right")) catch:  end try end if; if procname("left") <= _x0 then try procname(procname("left")) catch:  end try end if end if end if; return  elif type(_xin, `=`) and lhs(_xin) = "eventfired" then if not type(rhs(_xin), 'list') then error "'eventfired' must be specified as a list" end if; if _nv = 0 then error "this solution has no events" end if; if _dtbl[4] <> 2 and _dtbl[4] <> 3 then error "'direction' must be set prior to calling/setting 'eventfired'" end if; _i := _dtbl[4]; _val := NULL; if not assigned(_EnvEventRetriggerWarned) then _EnvEventRetriggerWarned := false end if; for _k in rhs(_xin) do if type(_k, 'integer') then _src := _k elif type(_k, 'integer' = 'anything') and type(evalf(rhs(_k)), 'numeric') then _k := lhs(_k) = evalf[max(Digits, 18)](rhs(_k)); _src := lhs(_k) else error "'eventfired' entry is not valid: %1", _k end if; if _src < 1 or round(_dtbl[1][3][1][_nv+1, 1]) < _src then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _src := {seq(`if`(_dtbl[1][3][1][_j, 1]-_src = 0., _j, NULL), _j = 1 .. _nv)}; if nops(_src) <> 1 then error "'eventfired' can only be set/queried for root-finding events and time/interval events" end if; _src := _src[1]; if _dtbl[1][3][1][_src, 2] <> 0. and _dtbl[1][3][1][_src, 2]-2. <> 0. then error "'eventfired' can only be set/queried for root-finding events and time/interval events" elif irem(round(_dtbl[1][3][1][_src, 4]), 2) = 1 then if _EnvEventRetriggerWarned = false then WARNING(`'eventfired' has no effect on events that retrigger`) end if; _EnvEventRetriggerWarned := true end if; if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then _val := _val, undefined elif type(_dtbl[_i][3][4][_src, _i-1], 'undefined') or _i = 2 and _dtbl[2][3][1][_src, 8] < _dtbl[2][3][4][_src, 1] or _i = 3 and _dtbl[3][3][4][_src, 2] < _dtbl[3][3][1][_src, 8] then _val := _val, _dtbl[_i][3][1][_src, 8] else _val := _val, _dtbl[_i][3][4][_src, _i-1] end if; if type(_k, `=`) then if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then error "cannot set event code for a rate hysteresis event" end if; userinfo(3, {'events', 'eventreset'}, `manual set event code `, _src, ` to value `, rhs(_k)); _dtbl[_i][3][1][_src, 8] := rhs(_k); _dtbl[_i][3][4][_src, _i-1] := rhs(_k) end if end do; return [_val] elif type(_xin, `=`) and lhs(_xin) = "direction" then if not member(rhs(_xin), {-1, 1, ':-left', ':-right'}) then error "'direction' must be specified as either '1' or 'right' (positive) or '-1' or 'left' (negative)" end if; _src := `if`(_dtbl[4] = 2, -1, `if`(_dtbl[4] = 3, 1, undefined)); _i := `if`(member(rhs(_xin), {1, ':-right'}), 3, 2); _dtbl[4] := _i; _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if; return _src elif _xin = "eventcount" then if _dtbl[1][3][1] = 0 or _dtbl[4] <> 2 and _dtbl[4] <> 3 then return 0 else return round(_dtbl[_dtbl[4]][3][1][_nv+1, 12]) end if else return "procname" end if end if; if _xout = _x0 then return [_x0, seq(evalf(_dtbl[1][6][_vmap[_i]]), _i = 1 .. _n-_ne)] end if; _i := `if`(_x0 <= _xout, 3, 2); if _xin = "last" and 0 < _dtbl[_i][4][9] and _dtbl[_i][4][9] < 100 then _dat := eval(_dtbl[_i], 2); _j := _dat[4][20]; return [_dat[11][_j, 0], seq(_dat[11][_j, _vmap[_i]], _i = 1 .. _n-_ne-_nd), seq(_dat[8][1][_vmap[_i]], _i = _n-_ne-_nd+1 .. _n-_ne)] end if; if not type(_dtbl[_i], 'array') then _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if end if; if _xin <> "last" then if 0 < 0 then if `dsolve/numeric/checkglobals`(op(_dtbl[1][14]), _pars, _n, _y0) then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars, _i) end if end if; if _dtbl[1][4][7] = 0 then error "parameters must be initialized before solution can be computed" end if end if; _dat := eval(_dtbl[_i], 2); _dtbl[4] := _i; try _src := `dsolve/numeric/SC/IVPrun`(_dat, _xout) catch: userinfo(2, `dsolve/debug`, print(`Exception in solnproc:`, [lastexception][2 .. -1])); error  end try; if _dat[17] <> _dtbl[1][17] then _dtbl[1][17] := _dat[17]; _dtbl[1][10] := _dat[10] end if; if _src = 0 and 100 < _dat[4][9] then _val := _dat[3][1][_nv+1, 8] else _val := _dat[11][_dat[4][20], 0] end if; if _src <> 0 or _dat[4][9] <= 0 then _dtbl[1][5][1] := _xout else _dtbl[1][5][1] := _val end if; if _i = 3 and _val < _xout then Rounding := -infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further right of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further right of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further right of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further right of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further right of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further right of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further right of %1", evalf[8](_val) end if elif _i = 2 and _xout < _val then Rounding := infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further left of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further left of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further left of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further left of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further left of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further left of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further left of %1", evalf[8](_val) end if end if; if _EnvInFsolve = true then _dig := _dat[4][26]; if type(_EnvDSNumericSaveDigits, 'posint') then _dat[4][26] := _EnvDSNumericSaveDigits else _dat[4][26] := Digits end if; _Env_dsolve_SC_native := true; if _dat[4][25] = 1 then _i := 1; _dat[4][25] := 2 else _i := _dat[4][25] end if; _val := `dsolve/numeric/SC/IVPval`(_dat, _xout, _src); _dat[4][25] := _i; _dat[4][26] := _dig; [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] else Digits := _dat[4][26]; _val := `dsolve/numeric/SC/IVPval`(eval(_dat, 2), _xout, _src); [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] end if end proc, (2) = Array(0..0, {}), (3) = [x, y(x), z(x)], (4) = []}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_rkf45, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_rkf45, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_rkf45, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_rkf45, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_rkf45), 'string') = rhs(x_rkf45); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_rkf45), 'string') = rhs(x_rkf45)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_rkf45) else _ndsol := 1; _ndsol := _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_rkf45) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error  end try end proc

 

 

 

 

 

Download odeSols.mw

The settings which you are trying to change *ought* to be controlled by your Maple.ini file.

For a single-user Windows installation the default location for this file is in the folder

C:\Users\USERID\AppData\Roaming\Maple\18

where the field USERID is specific to your installation.  Follow these steps.

  1. Check this file exists
  2. Assuming that this file exists in this location, copy/rename the file to the same folder - just call it MapleOld.ini
  3. Start Maple
  4. Make a simple change to your configuration settings, for example, toggle the Tools->Options->Display->Input Display setting from "2-D Math Notation" to "Maple Notation". Click "Apply Globally" and exit the Maple session.
  5. Load both the Maple.ini file and MapleOLD.ini file (created at stage 2 above) into a text editor and search for the field "Typeset Input" and verify that its value has been changed. "Typeset Input=false" corresponds to (old-fashioned) Maple Text input and "Typeset Input=true" corresponds to Maple 2-D Input

When I perform the above, comparison of the two "ini" files is as shown in the picture below

Now what happens when you try this????

 

 

are both shown in the attached, where I have also fixed the f[1,1] / f__11 issue highlighted by dharr)

I have no idea why you are using the ApproximateInt() command. If for some reason you want/have to employ this approach, then you should increase the number of sub-intervals for the Simpson method in order to obtain comparable numerical accuracy with the simple-minded approach.

The default number of sub-intervals for ApproximateInt(.... method=simpson...) is 10. As illustrated in the attached, this needs to be increased to ~100 in order to achieve numerical accuracy comparable with the "simple" method of performing the same integral

"restart;    `f__11`(r,theta,phi):=r^4 sin(6 theta) sin(3 phi):    L(r,theta,phi):=(2.784615385 10^10 ((&PartialD;)^2)/(&PartialD;r^2) `f__11`(r,theta,phi)+(2.784615385 10^10 (2+2 r cos(theta)) ((&PartialD;)/(&PartialD;r) `f__11`(r,theta,phi)))/(r (2+r cos(theta)))-(0.1175000000 (((&PartialD;)^4)/(&PartialD;theta^4) `f__11`(r,theta,phi)))/(r^4)-(0.1175000000 (((&PartialD;)^4)/(&PartialD;phi^4) `f__11`(r,theta,phi)))/((2+r cos(theta))^4)-(0.1175000000 (((&PartialD;)^4)/(&PartialD;phi^2&PartialD;r^2) `f__11`(r,theta,phi)))/((2+r cos(theta))^2)-(0.1175000000 (((&PartialD;)^4)/(&PartialD;r^2&PartialD;theta^2) `f__11`(r,theta,phi)))/(r^2)-(0.2350000000 (((&PartialD;)^4)/(&PartialD;phi^2&PartialD;theta^2) `f__11`(r,theta,phi)))/(r^2 (2+r cos(theta))^2)+(0.1175000000 ((cos(theta))^2 r^2+4 (cos(theta))^2 r^4+16 cos(theta) r^3-4+17 r^2) (((&PartialD;)^2)/(&PartialD;theta^2) `f__11`(r,theta,phi)))/((2+r cos(theta))^2 r^4)+(0.1175000000 (2 (cos(theta))^2 r^2+4 (cos(theta))^2 r^4+16 cos(theta) r^3+4+12 r^2) (((&PartialD;)^2)/(&PartialD;phi^2) `f__11`(r,theta,phi)))/(r^2 (2+r cos(theta))^4)-(2.784615385 10^10 (2 (cos(theta))^2 r^2+4 r cos(theta)+4) `f__11`(r,theta,phi))/(r^2 (2+r cos(theta))^2)+(0.2350000000 (((&PartialD;)^3)/(&PartialD;r&PartialD;theta^2) `f__11`(r,theta,phi)))/(r^3 (2+r cos(theta)))-(0.2350000000 (((&PartialD;)^3)/(&PartialD;phi^2&PartialD;r) `f__11`(r,theta,phi)))/(r (2+r cos(theta))^3)+(0.2350000000 sin(theta) (((&PartialD;)^3)/(&PartialD;theta^3) `f__11`(r,theta,phi)))/(r^3 (2+r cos(theta)))-(0.2350000000 sin(theta) (((&PartialD;)^3)/(&PartialD;phi^2&PartialD;theta) `f__11`(r,theta,phi)))/(r (2+r cos(theta))^3)+(0.1175000000 sin(theta) (((&PartialD;)^3)/(&PartialD;r^2&PartialD;theta) `f__11`(r,theta,phi)))/(r (2+r cos(theta)))-(0.1175000000 (2 r cos(theta)+3) sin(theta) (2 r cos(theta)+5) ((&PartialD;)/(&PartialD;theta) `f__11`(r,theta,phi)))/(r (2+r cos(theta))^3)) r^4 sin(6 theta) sin(3 phi):"

#
# Just do the integral the "simple" way
#
  Int( L(r, theta, phi),
       [r = 0.2..1, theta = 0..2*Pi, phi = 0..2*Pi]
     );
  evalf(%);

Int((0.3341538462e12*r^2*sin(6*theta)*sin(3*phi)+0.1113846154e12*(2+2*r*cos(theta))*r^2*sin(6*theta)*sin(3*phi)/(2+r*cos(theta))-101.5200000*sin(6*theta)*sin(3*phi)-9.517500000*r^4*sin(6*theta)*sin(3*phi)/(2+r*cos(theta))^4-63.45000000*r^2*sin(6*theta)*sin(3*phi)/(2+r*cos(theta))^2-4.230000000*(cos(theta)^2*r^2+4*cos(theta)^2*r^4+16*cos(theta)*r^3-4+17*r^2)*sin(6*theta)*sin(3*phi)/(2+r*cos(theta))^2-1.057500000*(2*cos(theta)^2*r^2+4*cos(theta)^2*r^4+16*cos(theta)*r^3+4+12*r^2)*r^2*sin(6*theta)*sin(3*phi)/(2+r*cos(theta))^4-0.2784615385e11*(2*cos(theta)^2*r^2+4*r*cos(theta)+4)*r^2*sin(6*theta)*sin(3*phi)/(2+r*cos(theta))^2-33.84000000*sin(6*theta)*sin(3*phi)/(2+r*cos(theta))+8.460000000*r^2*sin(6*theta)*sin(3*phi)/(2+r*cos(theta))^3-42.30000000*sin(theta)*r*cos(6*theta)*sin(3*phi)/(2+r*cos(theta))+12.69000000*sin(theta)*r^3*cos(6*theta)*sin(3*phi)/(2+r*cos(theta))^3-.7050000000*(2*r*cos(theta)+3)*sin(theta)*(2*r*cos(theta)+5)*r^3*cos(6*theta)*sin(3*phi)/(2+r*cos(theta))^3)*r^4*sin(6*theta)*sin(3*phi), [r = .2 .. 1, theta = 0 .. 2*Pi, phi = 0 .. 2*Pi])

 

0.5645200596e12

(1)

#
# Do the integral using approximateInt(), successively
# increasing the number of sub-intervals used in the
# Simpson's rule algorithm.
#
# The default is partition=10. Comparing output with
# the above "simple" method suggests that one needs
# to set partition=100 (approsimately) in order to
# obtain similar numerical accuracy
#
  with(Student[Calculus1]):
  for k from 10 to 100 by 10 do
      K1[k]:= evalf
              ( ApproximateInt
                ( L(r, theta, phi),
                  r = 0.2..1,
                  method = simpson,
                  output=sum,
                  partition=k
                )
              ):
      K2[k]:= evalf
              ( ApproximateInt
                ( K1[k],
                  theta = 0..2*Pi,
                  method = simpson,
                  output=sum,
                  partition=k
                )
              );
      K3[k]:= evalf
              ( ApproximateInt
                ( K2[k],
                  phi = 0..2*Pi,
                  method = simpson,
                  output=value,
                  partition=k
                )
              );
  od:
  K3();

(table( [( 10 ) = 0.5601317190e12, ( 20 ) = 0.5645182670e12, ( 30 ) = 0.5645201256e12, ( 40 ) = 0.5645200803e12, ( 50 ) = 0.5645200687e12, ( 60 ) = 0.5645200637e12, ( 70 ) = 0.5645200624e12, ( 90 ) = 0.5645200609e12, ( 80 ) = 0.5645200612e12, ( 100 ) = 0.5645200599e12 ] ))()

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Download myInt.mw

Sometimes it is quite a good idea to use packages provided by Maple for performing certain types of calculations.

For this particular problem, Maple's "geometry" package would seem to be appropriate.

See the attached

  restart;
  with(geometry):
#
# Define a line L through the points A and B
#
  line( L,
        [ point( A, [0, -3] ),
          point( B, [3,  1] )
        ]
      ):
#
# Compute the point Q which is the projection
# of the point P onto the line L
#
  projection( Q,
              point( P, [5, -2] ),
              L
            ):
#
# Get the coordinates of the desired point Q
#
  coordinates(Q);

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#
# Make the above calculation a procedure
# which will accept any three points
#
  getCoor:= proc( pt1::list, pt2::list, pt3::list)
                  uses geometry;
                  coordinates
                  ( projection
                    ( Q,
                      point( P, pt3),
                      line( l, [ point( A, pt1),
                                 point( B, pt2)
                               ]
                          )
                    )
                  );
             end proc:
  getCoor( [0, -3], [3, 1], [5, -2] );

[57/25, 1/25]

(2)

 


 

Download projPoint.mw

but my original comment about not really understanding teh problem still stands.

However at least this answer is vaguely(?) appropriate for this question

restart:
Severity:=[1,2,3]:
Likelihood:=[1,2,3]:
Impact:=LinearAlgebra:-OuterProductMatrix(Severity, Likelihood);
Outcome:=Matrix( op(1, Impact),
                 (i,j)-> if Impact[i,j] <= 3
                         then "minor"
                         elif Impact[i,j] <= 6
                         then "moderate"
                         else "serious"
                         fi
               );

Matrix(3, 3, {(1, 1) = 1, (1, 2) = 2, (1, 3) = 3, (2, 1) = 2, (2, 2) = 4, (2, 3) = 6, (3, 1) = 3, (3, 2) = 6, (3, 3) = 9})

 

Matrix(%id = 18446744074381287422)

(1)

 


 

Download outcomeMatrix.mw

 

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